Poisson Distribution Approximated by Hat-Check Distribution

Theorem

Let $X$ be a discrete random variable which has the hat-check distribution with parameter $n$.

Then $X$ can be approximated by a Poisson distribution with parameter $\lambda = 1$.

Proof

Let $X$ be as described.

Let $k \ge 0$ be fixed.

Then:

\(\ds \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - k }!} \sum_{s \mathop = 0}^k \dfrac {\paren {-1}^s} {s!}\)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \dfrac 1 {\paren {n - \paren {n - k} }!} \sum_{s \mathop = 0}^{n - k} \dfrac {\paren {-1}^s} {s!}\)

setting $k = n - k$

\(\ds \)

\(=\)

\(\ds \frac {1^k} {k!} e^{-1}\)

Taylor Series Expansion for Exponential Function and Definition of Poisson Distribution

Hence the result.

$\blacksquare$

Examples

Example: $N = 8$

Let $X$ be a discrete random variable which has the hat-check distribution with parameter $n = 8$.

Then $X$ can be approximated by a Poisson distribution with parameter $\lambda = 1$.